4,293 research outputs found
Probabilistic Spectral Sparsification In Sublinear Time
In this paper, we introduce a variant of spectral sparsification, called
probabilistic -spectral sparsification. Roughly speaking,
it preserves the cut value of any cut with an
multiplicative error and a additive error. We show how
to produce a probabilistic -spectral sparsifier with
edges in time
time for unweighted undirected graph. This gives fastest known sub-linear time
algorithms for different cut problems on unweighted undirected graph such as
- An time -approximation
algorithm for the sparsest cut problem and the balanced separator problem.
- A time approximation minimum s-t cut algorithm
with an additive error
Minimum Cuts in Near-Linear Time
We significantly improve known time bounds for solving the minimum cut
problem on undirected graphs. We use a ``semi-duality'' between minimum cuts
and maximum spanning tree packings combined with our previously developed
random sampling techniques. We give a randomized algorithm that finds a minimum
cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We
also give a simpler randomized algorithm that finds all minimum cuts with high
probability in O(n^2 log n) time. This variant has an optimal RNC
parallelization. Both variants improve on the previous best time bound of O(n^2
log^3 n). Other applications of the tree-packing approach are new, nearly tight
bounds on the number of near minimum cuts a graph may have and a new data
structure for representing them in a space-efficient manner
Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs
We introduce a new approach to computing an approximately maximum s-t flow in
a capacitated, undirected graph. This flow is computed by solving a sequence of
electrical flow problems. Each electrical flow is given by the solution of a
system of linear equations in a Laplacian matrix, and thus may be approximately
computed in nearly-linear time.
Using this approach, we develop the fastest known algorithm for computing
approximately maximum s-t flows. For a graph having n vertices and m edges, our
algorithm computes a (1-\epsilon)-approximately maximum s-t flow in time
\tilde{O}(mn^{1/3} \epsilon^{-11/3}). A dual version of our approach computes a
(1+\epsilon)-approximately minimum s-t cut in time
\tilde{O}(m+n^{4/3}\eps^{-8/3}), which is the fastest known algorithm for this
problem as well. Previously, the best dependence on m and n was achieved by the
algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute
approximately maximum s-t flows in time \tilde{O}(m\sqrt{n}\epsilon^{-1}), and
approximately minimum s-t cuts in time \tilde{O}(m+n^{3/2}\epsilon^{-3})
Graph Sample and Hold: A Framework for Big-Graph Analytics
Sampling is a standard approach in big-graph analytics; the goal is to
efficiently estimate the graph properties by consulting a sample of the whole
population. A perfect sample is assumed to mirror every property of the whole
population. Unfortunately, such a perfect sample is hard to collect in complex
populations such as graphs (e.g. web graphs, social networks etc), where an
underlying network connects the units of the population. Therefore, a good
sample will be representative in the sense that graph properties of interest
can be estimated with a known degree of accuracy. While previous work focused
particularly on sampling schemes used to estimate certain graph properties
(e.g. triangle count), much less is known for the case when we need to estimate
various graph properties with the same sampling scheme. In this paper, we
propose a generic stream sampling framework for big-graph analytics, called
Graph Sample and Hold (gSH). To begin, the proposed framework samples from
massive graphs sequentially in a single pass, one edge at a time, while
maintaining a small state. We then show how to produce unbiased estimators for
various graph properties from the sample. Given that the graph analysis
algorithms will run on a sample instead of the whole population, the runtime
complexity of these algorithm is kept under control. Moreover, given that the
estimators of graph properties are unbiased, the approximation error is kept
under control. Finally, we show the performance of the proposed framework (gSH)
on various types of graphs, such as social graphs, among others
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